We study the topological properties of random closed curves on the orientable surface $S$ of negative Euler characteristic. Let $\gamma_{n}$ denote the conjugate class of $n^{th}$ steps of a simple random walk on the Cayley graph driven by a non-elementary measure on the standard generating set, with probability converging to $1 When $ $n goes to infinity, (1) the length of $\gamma_{n}$ is minimized. (2) the number of self-intersections of $\gamma_{n}$ is on the order of $n^{2}$ and (3) $S$ is punctured and the distribution is If uniform, if the minimum order $\gamma_{n}$ of the cover allows a simple lift (called $\textit{simple lift}$ of $\gamma_{n}$), then at least $ It grows as n/\log(n)$ and is on the order of $n$ at most.

It also shows that these properties are $\textit{generic}$. This means that the proportion of elements of spheres of radius $n$ in the Cayley graph they hold converges to $1$ as $n$ goes. infinite.

The resulting simple lifting bound for the randomly chosen curve significantly improves the previously known best range, which was of the order of $\log^{(1/3)}n$. increase. As an application, we give the extension of the general point-push pseudo-Anosov homeomorphism relatively sharp upper and lower bounds on the number of self-intersections of its defining curve, and also give an upper bound on the random simple lifting order. A curve with a better number of points of intersection than the general curve boundary.

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